Here \(A_{\chi }=2\, E\, V_{\chi }\)  and \(V_{\chi }\)  is the dark matter potential similar to \(V_{CC}\)  in the case of ordinary matter. On the other hand if DM interacts only with muon-neutrinos [1]}, the form of \(\mathcal {H}_{\chi }\)  changes to
\(\mathcal {H_{\chi }}=\frac{1}{2E}\begin{pmatrix}0 & 0 & 0\\0 & A_{\chi } & 0\\0 & 0 & 0\end{pmatrix}\,.\) 
